The area bounded by the lines ` y= || x-1| -2| ` and `x` axis is

Answer

To find the area bounded by the curve \( y = ||x - 1| - 2| \) and the x-axis, we will follow these steps:

### Step 1: Understand the function
The function \( y = ||x - 1| - 2| \) involves absolute values, which means we need to break it down into different cases based on the values of \( x \).

### Step 2: Break down the absolute values
1. **First absolute value**: \( |x - 1| \)
- For \( x < 1 \): \( |x - 1| = 1 - x \)
- For \( x \geq 1 \): \( |x - 1| = x - 1 \)

2. **Second absolute value**: \( ||x - 1| - 2| \)
- We need to consider the cases based on the first absolute value's output.

### Step 3: Analyze cases for \( ||x - 1| - 2| \)
1. **Case 1**: \( x < -1 \)
- \( |x - 1| = 1 - x \) → \( ||x - 1| - 2| = |(1 - x) - 2| = | - x - 1| = x + 1 \)

2. **Case 2**: \( -1 \leq x < 3 \)
- \( |x - 1| = 1 - x \) → \( ||x - 1| - 2| = |(1 - x) - 2| = | - x - 1| = x + 1 \)

3. **Case 3**: \( x \geq 3 \)
- \( |x - 1| = x - 1 \) → \( ||x - 1| - 2| = |(x - 1) - 2| = |x - 3| \)
- For \( x \geq 3 \): \( ||x - 1| - 2| = x - 3 \)

### Step 4: Find the points of intersection with the x-axis
To find the area bounded by the curve and the x-axis, we need to find where \( y = 0 \).

1. From the analysis:
- For \( x < -1 \): \( x + 1 = 0 \) → \( x = -1 \)
- For \( -1 \leq x < 3 \): \( x + 1 = 0 \) → \( x = -1 \) (already found)
- For \( x \geq 3 \): \( x - 3 = 0 \) → \( x = 3 \)

Thus, the points of intersection with the x-axis are \( x = -1 \) and \( x = 3 \).

### Step 5: Determine the area
The area bounded by the curve and the x-axis can be calculated by integrating the function from \( x = -1 \) to \( x = 3 \).

1. **Area Calculation**:
- For \( -1 \leq x < 3 \): \( y = x + 1 \)
- For \( x \geq 3 \): \( y = x - 3 \) (but this part does not contribute to the area since it is above the x-axis)

The area can be calculated as:
\[ \text{Area} = \int_{-1}^{3} (x + 1) \, dx \]

### Step 6: Calculate the integral
\[ \text{Area} = \left[ \frac{x^2}{2} + x \right]_{-1}^{3} \]
Calculating the definite integral:
\[ = \left( \frac{3^2}{2} + 3 \right) - \left( \frac{(-1)^2}{2} - 1 \right) \]
\[ = \left( \frac{9}{2} + 3 \right) - \left( \frac{1}{2} - 1 \right) \]
\[ = \left( \frac{9}{2} + \frac{6}{2} \right) - \left( \frac{1}{2} - \frac{2}{2} \right) \]
\[ = \frac{15}{2} - \left( -\frac{1}{2} \right) \]
\[ = \frac{15}{2} + \frac{1}{2} = \frac{16}{2} = 8 \]

### Final Answer
The area bounded by the curve and the x-axis is \( 8 \).

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